Here is a short biography of Euler, which mentions that he introduced the beta and gamma functions, as integrals, in 1729, and that they were named by later mathematicians. In particular:

He also studied beta and gamma functions, which he had introduced in 1729. Legendre called these ‘Eulerian integrals of the first and second kind’ respectively while they were given the names beta function and gamma function by Binet and Gauss respectively.

One use of the beta function is as the normalization factor in the beta distribution. It also arises in the stochastic urn process known as a preferential attachment process.
One might also read about physicist Gabriele Veneziano, who, in applying the beta function to the scattering amplitudes of particles interacting via the strong force, led to the creation of string theory.
I hope these give you places to start